Supabase is a dockerised postgres with user auth, rest API and some other goodies. It’s maybe too complicated as a starter.
Appwrite might also work for ya. Much easier to get into, but also less feature complete.
Pocketbase might also work. Haven’t used it tho
Nginx Proxy Manager is probably perfect for you.
Pick a domain (like mylab.home or something), set up your home network to resolve that domains IP as your docker hosts IP.
NPM will do self-signed certs. So, you will get a “warning, Https is insecure” kinda page when you visit it. You could import NPMs root cert into your OS/browser so it trusts it (or set up an “don’t warn for this domain” or something).
If you don’t want per-client config to trust it, then you need to buy a domain, use a DNS that supports letsencrypt DNS-challenge, and grab certs that way (means you don’t need a publicly accessible well-known route exposed)